New approach to weighted topological entropy and pressure

TL;DR

This paper introduces a novel approach to weighted topological entropy and pressure, providing new definitions and establishing their equivalence, which generalizes the dimension formula for certain fractal sets in a purely topological framework.

Contribution

It presents a new methodology for defining weighted topological entropy and pressure, differing from previous definitions, and proves their equivalence, extending the dimension formula for fractal carpets.

Findings

New definitions of weighted topological entropy and pressure

Proved the equivalence of the new and original definitions

Generalized the dimension formula for Bedford--McMullen carpets

Abstract

Motivated by fractal geometry of self-affine carpets and sponges, Feng--Huang (2016) introduced weighted topological entropy and pressure for factor maps between dynamical systems, and proved variational principles for them. We introduce a new approach to this theory. Our new definitions of weighted topological entropy and pressure are very different from the original definitions of Feng--Huang. The equivalence of the two definitions seems highly nontrivial. Their equivalence can be seen as a generalization of the dimension formula for the Bedford--McMullen carpet in purely topological terms.